Further evidence for near-Tsirelson Bell–CHSH violations in quantum field theory via Haar wavelets

Abstract This paper investigates a recent construction using bumpified Haar wavelets to demonstrate explicit violations of the Bell–Clauser–Horne–Shimony–Holt inequality within the vacuum state in Quantum Field Theory. The construction was tested for massless spinor fields in $$(1+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional Minkowski spacetime and is claimed to achieve violations arbitrarily close to an upper bound known as Tsirelson’s bound. We show that this claim may be reduced to a mathematical conjecture involving the maximal eigenvalue of a sequence of symmetric matrices composed of integrals of Haar wavelet products. More precisely, the asymptotic eigenvalue of this sequence should approach $$\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> . We present a formal argument using a subclass of wavelets, allowing to reach 3.11052. Although a complete proof remains elusive, we present further compelling numerical evidence to support it.